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Zbl 0833.60060
Cohen, S.
Some Markov properties of stochastic differential equations with jumps. (English)
[A] Azéma, J. (ed.) et al., Séminaire de probabilités XXIX. Berlin: Springer-Verlag. Lect. Notes Math. 1613, 181-193 (1995). ISBN 3-540-60219-4/pbk

In previous articles we were interested in SDE's on manifolds driven by non-continuous semimartingales; we got an extension of Meyer-Schwartz's second order calculus. Here we will be concerned with Markov properties of solutions, when the driving process is a Lévy process living in a vector space, more precisely we will compute their infinitesimal generator. As an application we are studying a process called pseudo- alpha-stable process constructed as the stochastic development on a Riemannian manifold of a vector-valued alpha-stable process living in the tangent space of the manifold. We compare this pseudo-alpha-stable process with the alpha-stable process on the manifold obtained as a Brownian motion time-changed by a suitable subordinator. We give a probabilistic proof that pseudo-alpha-stable and alpha-stable processes do not have the same laws on suitable Riemannian manifolds with a pole and a rotationally invariant metric. On the sphere in dimension 2, it is worth mentioning that the infinitesimal generator of Brownian motion and of the pseudo-1-stable process are linked via a formula involving a concave, piecewise affine function. As a consequence of this remark the pseudo-1-stable process on the 2-sphere is not a Brownian motion time- changed by a subordinator.
[S.Cohen (Noisy le Grand)]

MSC 2000:: *60H10 Stochastic ordinary differential equations
60J70 Appl. of diffusion theory

Keywords: stochastic analysis on manifolds; Markov processes; jump processes; Meyer-Schwartz's second order calculus; Riemannian manifold; Brownian motion

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